\section{IBM Model-1}
\label{sec:ibmModel1}

IBM Model I is a word-based translation model that utilizes EM to calculate
lexical translation probabilities given two parallel corpora. Give a language
model $p(e)$ and a translation model $p(f|e)$, an English translation $e^*$ of French source
sentence $f$ can be found following Bayes' rule:
\begin{equation}
e^* = \arg\max_{e \in E} P(e|f) = \arg\max_{e \in E} p(e) \times P(f|e)
\end{equation}
In order to calculate $P(f|e)$, IBM Model I makes use of so called alignment
variables:
\begin{equation}
p(f|e) = \sum\limits_a p(f, a|e)
\end{equation}

\begin{equation}
p(a|f,e) = \frac{p(f,a|e)}{\sum\limits_{a'} p(f, a'|e)}
\end{equation}


The \emph{Viterbi alignments} are the alignments that are the most probable for each given sentence pair.
